Color not showing up and neither is the option
Weight Painting, Texture Painting and Vertex Painting are all completely different.

Weight painting directly modifies vertex groups, which can be used to designate areas of an object (such as where you want hair to grow).

Vertex painting allows you to color the individual vertices. This technically could work, however if you have a low-poly model it's going to look pretty bad. Additionally, as far as I'm aware, vertex painting is not compatible with Cycles. PGMath points out that it is possible with Attribute nodes.

Texture painting is what you want.



Exhibit A: Our model

Step 1: UV Unwrap Your Model

If you have already done this, or already know how, go ahead and skip this step.

Basically you want to go into edit mode and select a bunch of vertices so that if you took scissors and cut along the selected lines, you'd end up with a bunch of more or less flat objects. Like so:



Now press CTRL + E and click Mark Seam.



Press A twice so that every vertex is selected, and then press U and click Unwrap.



Great! You're done with this step.

Step 2: Setting up the texture

If you have already done this, or know how, you can skip this step too.

First you need to create a texture for your model. Making sure your model is selected, and that it has a material, go to the texture panel and click + New.



Under Image, click + New and use whatever settings you'd like.




  @cegaton says:
  
  Beware: blender does not automatically save the texture in the blend
  file. The texture has to be saved first by pressing CTRL F3 or using
  Image Save in the image editor. Otherwise the next time you open the
  file all of your texturing work will be gone.


Step 3: Setting up the texture painting

Almost done! Go into Texture Paint mode. Press T to make the panel on the left visible. Click on the Slots tab, set the Painting Mode to Image, and select the image you just created.



Now you should be good to go!

Step 4: But it still doesn't work!

-- Make sure you don't have a texture set

Though it's not very clear, the Texture setting does not set the texture it draws on. It sets the texture that the brush uses. Unless you have a specific purpose for using it, it should not be set. If your texture panel looks like this:



That is probably your problem. Click the little X to the right of the F and +.

-- Make sure your settings are configured correctly

These settings may cause you problems:

Drawing mode: You want TexDraw. That should be the default.

Foreground color: If it's the same as the background color of the image, it won't have any effect obviously.

Strength: Make sure it's larger than zero.

Blend: Set it to Mix to make sure it's working. You can mess with it later.



The End



Blender has so many features that a problem like this could be any number of things. If you've followed all the steps and it still didn't work, I'm stumped. If you could upload your .blend file though, we would have a much easier time helping you.